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Michel Theory of Symmetry Breaking and Gauge Theories

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Michel Theory of Symmetry Breaking and Gauge Theories Michel theory of symmetry breaking and gauge theories Giuseppe Gaeta∗ I.H.E.S., 35 Route de Chartres 91440 Bures sur Yvette (France) [email protected] Paola Morando Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino (Italy) [email protected] Summary. We extend Michel’s theorem on the geometry of symmetry breaking [1] to arXiv:math-ph/0207002v1 29 Jun 2002 the case of pure gauge theories, i.e. of gauge-invariant functionals defined on the space C of connections of a principal fiber bundle. Our proof follows closely the original one by Michel, using several known results on the geometry of C. The result (and proof) is also extended to the case of gauge theories with matter fields. ∗ On leave from Dept. of Mathematics, Loughborough University, Loughborough LE11 3TU (GB) 27/1/97 Research supported by the Volkswagen Stiftung under the RiP program at Oberwolfach. Introduction In 1971, Louis Michel – motivated by the SU(3) theory of hadronic interactions – proved a remarkable theorem on symmetry breaking in theories described by a G-invariant potential (G a compact semisimple Lie group) over a finite dimensional smooth G-manifold M [1]; this result was a direct generalization of the theory he and Radicati had developed to study the geometry of the SU(3) octet, and the model-independent features of the SU(3) theory [2-4]. Essentially, Michel theorem guarantees that, under suitable conditions, there are points in M which are critical for any G-invariant potential V : M → R; moreover, these points are characterized in terms of a geometric construction which takes into account the symmetry properties of points – and subsets – of M under the G action. Notice that actually, critical points of G-invariant functions come necessarily in G-orbits; thus, whenever the orbit space Ω = M/G is well defined – as it is under the conditions mentioned above – it is convenient to set the problem directly in Ω (maybe with an explict use of the basis of G-invariant functions on M as coordinates in Ω), as Michel did. Thus, in summary, the Michel theorem on critical orbits of G-invariant functions on a G-manifold [1] allows to identify G-orbits which are critical for any G-invariant function; thus, it permits to study spontaneous symmetry breaking in a model-independent way [3-16]; this theory has also been extended to study supersymmetry breaking [12]. Mathematical foundations for the theory are provided e.g. by [17,18]. The purpose of the present note is to show that Michel’s theory can be extended to the study of (pure) gauge theories, such as those defined by a Yang-Mills functional, and more generally by a gauge-invariant functional defined on a space of connections for a given principal bundle. It will turn out that once this is obtained, extension to the case of complete gauge theories, i.e. theories with matter fields, follows rather easily1. It should be stressed that, for our result to be of physical interest, the space of connections to be considered (see below) should obey some natural conditions. Thus, on the one side we would be not justified in restricting our attention to the (dense) subspace of irreducible connections, for which a full geometric description is well known [22,23], and which is relatively simple (we recall however that in the SU(2) case this coincides with the full space of connections [24]). On the other side, we should not consider the full space of connections, but only those connections satisfying a finite energy condition, or 1 In this respect, it should be mentioned that an extension of Michel theory to gauge func- tionals (in that case, dealing only with matter fields) was already considered in [19-21], but this suffered from several – strong – limitations; these followed from focusing on the seemingly easier sector of matter fields, and it turns out that the present approach is not only more natural geometrically, but also more fruitful. 1 more precisely the completion of this set in suitable norm; this leads naturally to consider Sobolev spaces of connections and Sobolev norms [24,25]. The same remark would apply when considering sections of a vector bundle in the context of gauge theories with matter fields, see below. In the present note, we will give a careful statement and a complete proof of the extension of Michel’s theorem to pure gauge theories; we will also shortly discuss some – easy but relevant – extensions and generalizations of this (including the one to theories with matter fields), whose proof is only sketched or omitted at all as it would just repeat the one for the pure gauge case, and some of the possible applications. It is appropriate to stress here that it turns out – as it will be clear in our discussion – that one can directly extend the essence of Michel’s construction and proof, from its original setting – M a finite dimensional manifold, G a compact Lie group, in the following referred to as the “classical case” – to the case of gauge theories, which involve infinite dimensional, non-compact manifold and group (a large part of the results needed for this are classical ones [22,23,26]); this shows (see also the final discussion in sect.7) how far-reaching is the Michel theory of symmetry breaking. Plan of the paper. Let us shortly describe the pan of the paper. We will start by collecting in sect.1 some well known geometrical fact to provide a proper setting – and fix notation – for the discussion to follow. In sect.2 we will define a Sobolev norm on the space of connections and on gauge sections, and choose some Sobolev class of connections – and correspondingly of gauge sections – in which we work; this is needed in order to deal with Hilbert spaces. With this, we can use known results concerning existence of slices. The central result of this note is given in sect.3; it is a direct extension of Michel’s theorem [1] – and proof – to the pure gauge case; after this, we can pass to consider extension and consequences of it. Thus, first of all we notice in sect.4 that once we have the result for the pure gauge case, it can be easily extended to the case of gauge theories with matter fields. Then, in sect.5 we discuss some other extensions, dealing with general systems (i.e. not necessarily variational ones) with a gauge symmetry, and we formulate the gauge equivalent of reduction lemmata which are well known in the finite dimensional case; these are applied in sect.6 to discuss specifically the problem of symmetry breaking in theories with a (continuous) dependence on a parameter. Finally, in sect.7 we discuss some further extensions, as well as the strength – and weakness – of the approach proposed here. The discussion of sects.5-7 refers only to the pure gauge case for ease of notation, but it will be clear from sect.4 that these would also extend to the case where matter fields are present. All our discussion will be conducted using only the geometry of the problem, i.e. with no reference to the kind of specific functional one encounters in physical examples of gauge theories. To discuss these is obviously a very relevant matter, so that we do indeed consider them, and how our main theorem can be of use in determining specific solutions, in the appendices. Here we restrict specifically to the well known class of functionals made of 2 the Yang-Mills one L = |Fµν | for the pure gauge part, and by the standard Lagrangian L =[|∇φ|2 − V (φ)] (see below) for the matter fields (with ∇ the covariant derivative). In appendix A we study of special class of solutions, in appendix B we further restrict this 2 to the case associated to the null connection A0 (see below), and finally in appendix C we deal with the case G = SU(3). 1. Geometrical setting Let B be an n-dimensional (n finite) riemannian manifold (e.g. B = R4 or S4); let G be a compact, connected and semisimple Lie group (e.g. G = SU(N)); let P be a principal fiber bundle over B with projection π : P → B and fiber π−1(x) ≈ G. We denote by G the Lie algebra of G. Let C be the set of connections on P . It is well known that C is an affine space modelled on the vector space A := Λ1 (T∗B, G) of the one-forms on B taking value in G. With any connection A ∈ C we can associate a covariant derivative ∇α and a one-form α ∈ A that 1 n are given, in local coordinates (x ,...,x ) on B and with Aµ : B → G, by µ α = Aµ(x) dx ; (1) α ∇µ = ∂µ + Aµ . (2) 0 0 Choosing as reference point in C the null connection A such that Aµ =0(µ = 1,...,n), from now on we will say “the connection α” to mean “the connection A = A0 + α whose associated connection form is α”. Let Γ denote the space of differentiable sections of P ; Γ has the structure of a Hilbert-Lie group, and a section γ ∈ Γ will be written in local coordinates as γ = g(x) , (3) with g : B → G. We say that Γ is a gauge group modelled on G. The gauge group Γ acts naturally on the operator of covariant derivative associated to a connection A ∈ C by conjugation, i.e. γ : ∇α → γ∇αγ−1; the action of Γ on A is given in local coordinates by −1 −1 γ : Aµ =⇒ g(x) · Aµ(x) · g (x) − (∂µg)(x) · g (x) .
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